Numerical investigations of SO(4) emergent extended symmetry in spin-$\frac{1}{2}$ Heisenberg antiferromagnetic chains

The antiferromagnetic Heisenberg chain is expected to have an extended symmetry, $[\mathrm{SU}\left(2\right)\times \mathrm{SU}\left(2\right)]/{\mathrm{Z}}_2$, in the infrared limit whose physical interpretation is that the spin and dimer order parameters form the components of a common four-dimensional pseudovector. Here we numerically investigate this emergent symmetry using quantum Monte Carlo simulations of a modified Heisenberg chain (the $J\text{−}Q$ model) in which the logarithmic scaling corrections of the conventional Heisenberg chain can be avoided. We show how the two- and three-point spin and dimer correlation functions approach their forms constrained by conformal field theory as the system size increases and numerically confirm the expected effects of the extended symmetry on various correlation functions. We stress that sometimes the leading power laws of three-point (and higher) correlations are not given simply by the scaling dimensions of the lattice operators involved but can be faster decaying because of exact cancellations of contributions from the fields and currents under conformal symmetry.

Figure 1

The scaled three-point function for the critical periodic TFIM chain as defined in Eq. (21) flows to a constant for $x=L/8$ and $x=L/2$. The line is a guide to the eye, and for $x=L/2$, we exclude data only around $y/L=0.5$ for large sizes because the correlation function vanishes quickly as $x\sim y\sim L/2$, which means the numerical signal is very weak.

Figure 2

Three-point function for the critical TFIM periodic chain scaled using the lattice distance rather than the conformal distance for $x=L/8$.

Figure 3

Scaled dimer two-point function for the critical $J{Q}_2$ periodic chain as defined in Eq. (31). The results flow to a constant with increasing size. The horizontal axis extends to only $x/L=0.5$ as two-point functions are symmetric about $y=L/2$.

Figure 4

Spin two-point function for the critical $J{Q}_2$ chain scaled with the first term of Eq. (28) [as explicitly shown in Eq. (32)]. The approach to a constant for large sizes confirms the expected form.

Figure 5

Spin two-point function for the critical $J{Q}_2$ chain with the first term of Eq. (28) subtracted out and scaled with the second term, as in Eq. (33). We see improving agreement with the expected form with increasing $L$.

Figure 6

The Dimer three-point function for the critical $J{Q}_2$ chain vanishes for large enough system size, which agrees with Eq. (26), shown here for $x=L/4$ and $x=L/2$.

Figure 7

The scaled dimer-spin-spin three-point function, Eq. (34), for the critical $J{Q}_2$ flows to a constant, demonstrated for $x=L/4$ and $x=L/2$. In the top panel, the divergence at $y=3L/4$ is due to the exact vanishing of the correlation function at this point. In the bottom panel, data for even $y$ values in the range $y/L\in (0.3,0.7)$ have been excluded as $\langle {\mathbb{D}}_0{\vec{\mathbb{S}}}_x·{\vec{\mathbb{S}}}_y\rangle$ tends to a “0/0” form, and thus, the scaled correlation function is very noisy.